Quarks are spin-1/2 fields for which a Dirac operator can be studied. The local boundary conditions proposed for models of quark confinement are naturally related to the local boundary conditions studied, more recently, in one-loop quantum cosmology. Further developments lie in the possibility of studying quantization schemes in conformally invariant gauges. This possibility is investigated in the case of the Eastwood–Singer gauge for vacuum Maxwell theory on manifolds with boundary. This is part of a more general scheme, leading to the analysis of conformally covariant operators. These are also presented, with emphasis on the Paneitz operator. In spectral geometry, a class of boundary operators are described which include the effect of tangential derivatives. They lead to many new invariants in the heat-kernel asymptotics for operators of Laplace type. The consideration of tangential derivatives arises naturally within the framework of recent attempts to obtain Becchi–Rouet–Stora–Tyutin-invariant boundary conditions in quantum field theory. However, in Euclidean quantum gravity, it remains unclear how to write even just the general form of the various heat-kernel coefficients. Last,the role of the Dirac operator in the derivation of the Seiberg– Witten equations is described. The properties of the new scheme, with emphasis on the invariants and on the attempts to classify four-manifolds, are briefly introduced.

Introduction

So far we have dealt with many aspects of manifolds with boundary in mathematics and physics. Hence it seems appropriate to begin the last chapter of our monograph with a brief review of the areas of research which provide the main motivations for similar investigations. They are as follows.